GRB/GW ASSOCIATION: LONG–SHORT GRB CANDIDATES, TIME LAG, MEASURING GRAVITATIONAL WAVE VELOCITY, AND TESTING EINSTEIN’S EQUIVALENCE PRINCIPLE

Xiang Li (李翔); Yi-Ming Hu (胡一鸣); Yi-Zhong Fan (范一中); Da-Ming Wei (韦大明)

doi:10.3847/0004-637X/827/1/75

1. INTRODUCTION

The coalescence of a binary compact object system (either a neutron star (NS) binary or a stellar-mass black hole (BH) and NS binary) has been widely suggested to account for short-duration gamma-ray burst (SGRB) events (Eichler et al. 1989; Narayan et al. 1992; Nakar 2007; Berger 2014) that last typically less than 2 s in the soft γ–ray band (Kouveliotou et al. 1993). Since 2006, it has been suspected that mergers of compact objects could also produce the so-called long–short GRBs (also known as the supernova-less long GRBs, which are apparently long-lasting but do not show any signal of supernovae down to very stringent limits), which share some properties of both long- and short-duration GRBs (Della Valle et al. 2006; Gal-Yam et al. 2006; Gehrels et al. 2006; Zhang et al. 2007). Compact binary coalescence (CBC) is generally expected to be a strong source of gravitational wave (GW) radiation and such events are prime targets for some GW detectors like advanced Laser Interferometer Gravitational-Wave Observatory (LIGO)/VIRGO (Abadie et al. 2015; Acernese et al. 2015; Belczynski et al. 2010, 2016; see also the latest LSC–Virgo white paper at https://dcc.ligo.org/LIGO-T1400054/public). On 2015 September 14, the two detectors of the LIGO simultaneously detected a transient gravitational wave signal from the merger of two BHs (GW150914, Abbott et al. 2016). GW150914 is the first direct detection of gravitational waves and the first identification of a binary BH merger (Abbott et al. 2016). Surprisingly, the observations from the Fermi Gamma-ray Burst Monitor (GBM) at the time of GW150914 claimed a detection of a weak gamma-ray transient (i.e., GBM transient 150914) 0.4 s after GW150914 with a false alarm probability of 0.0022 (Connaughton et al. 2016). If true, this is the first GW/SGRB association (see, however, Savchenko et al. 2016 for some arguments). Li et al. (2016) compared GBM transient 150914 with other SGRBs and found that such an event is remarkably different in its prompt emission properties. The binary BH merger origin as well as its property of "distinguished" prompt emission suggest that GW150914/GBM transient 150914 is not a typical GW/SGRB association.

In the absence of successful detection of the gravitational radiation triggered by a "normal" (S)GRB (Abadie et al. 2012; Aasi et al. 2014a, 2014b), a "smoking-gun" signature for a compact-binary origin would be the detection of the so-called Li–Paczynski macronova (also called a kilonova), which is a near-infrared/optical transient powered by the radioactive decay of r-process material synthesized in the ejecta that is launched during the merger event (e.g., Li & Paczyński 1998; Kulkarni 2005; Metzger et al. 2010; Barnes & Kasen 2013). The identifications of macronova candidates in the afterglows of the canonical short event GRB 130603B (Tanvir et al. 2013; Berger et al. 2013), the long–short burst GRB 060614 (Jin et al. 2015; Yang et al. 2015), and the short event with extended X-ray emission GRB 050709 (Jin et al. 2016) strongly support the CBC origin of some GRBs. A conservative estimate of the macronova rate suggests that the prospects for detection of the GW radiation by the (upcoming) advanced LIGO detectors are promisiing (Jin et al. 2015). We anticipate that in the near future many GW sources driven by the merger of compact objects would be detected (Abadie et al. 2010) and a small fraction of such events would be accompanied by supernova-less GRBs (including both the short and long–short events).

The observation of a "nearby" supernova-less GRB provides a reliable estimate of the time, sky location, and distance of a potential binary merger signal. This significantly reduces the parameter space of a follow-up GW search and consequently could be used to reduce the effective detection threshold and effectively increase the detectors' sensitivity and their detection rate (e.g., Kochanek & Piran 1993; Finn et al. 1999; Harry & Fairhurst 2011; Dietz et al. 2013; Kelley et al. 2013; Nissanke et al. 2013; Williamson et al. 2014; Bartos & Marka 2015; Clark et al. 2015). In this work we examine whether some SGRBs and/or long–short GRBs are within the horizon of the advanced LIGO/VIRGO network and discuss the model-dependent time lag between the coalescence and GRBs.

This work is structured as follows: in Section 2 we discuss/summarize the prospect of detecting GW-associated GRBs in the era of advanced LIGO/VIRGO and examine whether any recent GRBs (either SGRBs or long–short GRBs) are within the horizon of the advanced LIGO/VIRGO network. In Section 3 the model-dependent time lags between the merger and GRBs are presented and the possibility of revealing the nature of merger remnants with such time delays is discussed. The expected progress in measuring the speed of GWs with the future data is investigated in Section 4. Our results and discussion are presented in Section 5.

2. THE PROSPECTS OF DETECTING GW SIGNALS ASSOCIATED SGRBs AND LONG–SHORT GRBs

2.1. The Prospect of Detecting SGRBs Associated with GW Signals

The strategy of the targeted search for GWs associated with short GRBs has been discussed extensively in Harry & Fairhurst (2011) and Williamson et al. (2014). The prospect of detecting SGRBs with GW signals has also been widely estimated in the literature (e.g., Williamson et al. 2014; Clark et al. 2015; Wanderman & Piran 2015), and in this subsection we simply summarize their main conclusions.

2.1.1. Binary NS (BNS) Mergers

The sensitive distance can be approximated as (Clark et al. 2015)

$\begin{eqnarray}&&{D}_{* ,{\rm{BNS}}}\approx 400\,{\rm{Mpc}}\,(9/{\rho }_{* }),\end{eqnarray} \tag{ 1 }$

where ${\rho }_{* }$ is the signal-to-noise ratio of the GW signal. When the advanced LIGO/VIRGO (aLIGO/AdV) network has reached its full sensitivity, with a "local" SGRB detection rate of $4\pm 2\,{\mathrm{Gpc}}^{-3}\,{\mathrm{yr}}^{-1}$ (Wanderman & Piran 2015), the detection rate of SGRBs associated with a GW signal for a full-sky γ-ray monitor is estimated as

$\begin{eqnarray}&&{{ \mathcal R }}_{{\rm{full\; sky}}}({\rho }_{* }=9)\approx 1\pm 0.5\,{\mathrm{yr}}^{-1},\end{eqnarray} \tag{ 2 }$

where all SGRBs are assumed to originate from BNS mergers. Such an assumption seems reasonable since the BH–NS merger rate is generally expected to be just $\sim 1/10$ times that of the BNS merger rate (Abadie et al. 2010). Note that the LIGO/Virgo network can boost GW detection rates by exploiting the mass distribution of the neutron stars within the BNS system, and for searches with detected electromagnetic counterparts the detection rate may increase by 60% (Dent & Veitch 2014; Bartos & Marka 2015).

2.1.2. BH–NS Mergers

No neutron star–black hole (NS–BH) binaries have yet been observed directly (Lattimer 2012) but indirect evidence for NS–BH mergers has been suggested in the macronova modeling (Jin et al. 2015; Yang et al. 2015). In a systematic analysis of the BH mass distribution based on 35 X-ray binaries, Farr et al. (2011) found strong evidence for a mass gap between the most massive neutron stars and the least massive BHs, confirming the results of Bailyn et al. (1998) and Ozel et al. (2010). For the low-mass systems (combined sample of systems), they found a BH mass distribution whose 1% quantile lies above 4.3 ${M}_{\odot }$ (4.5 ${M}_{\odot }$ ) with 90% confidence. Typical NS–BH binary systems are expect to have a mass ratio of ~1:4, for which the sensitive distance of the aLIGO/AdV network can be estimated as

$\begin{eqnarray}&&{D}_{* ,\mathrm{NS}\mbox{--}\mathrm{BH}}\approx 690\,{\rm{Mpc}}\,(9/{\rho }_{* }).\end{eqnarray} \tag{ 3 }$

If $\sim 1/5$ of short and long–short GRBs are produced by NS–BH mergers, while aLIGO/AdV are more sensitive to the heavier NS–BH systems, we expect that around half of the CBC events might have an electromagnetic counterparts originating from NS–BH mergers.

As an optimistic estimate (i.e., supposing most nearby SGRBs can be observed by Fermi GBM-like detectors), in 10 years of full running of the aLIGO/AdV network ~10–20 GW-associated SGRBs are expected, and possibly one half of them may originate from NS–BH mergers. The statistical study of such a sample, though still limited, may shed valuable light on the physical processes taking place at the central engine (see Section 3 for details) and possibly also the fundamental physics (see Section 4 for details). In addition to SGRBs, some supernova-less long GRBs may also originate from a compact object merger and the detection rate of merger events will increase.

2.2. "Supernova-less" GRBs within the Sensitivity Distance of the Advanced LIGO/VIRGO Network

As shown in Section 2.1, the prospect of detecting GW-associated SGRBs is promising for the advanced LIGO/VIRGO network. The nearest short burst is GRB 061201. It is measured to have a redshift of z = 0.111 (Berger 2014; see, however, D'Avanzo et al. 2014 for the uncertainty), or a distance of 520 Mpc, which is larger than ${D}_{* ,{\rm{BNS}}}({\rho }_{* }=9)$ . Note that most SGRBs are expected to be powered by BNS mergers, hence no single SGRB has been found within the averaged sensitive distance of the advanced LIGO/VIRGO network (Williamson et al. 2014; Wanderman & Piran 2015). Such a result is somewhat disappointing though not in significant tension with the expectation (see Equation (2)). To better explore the situation, in this work we also take into account all "nearby" (i.e., $z\lt 0.3$ ) supernova-less long GRBs, including XRF 040701 (X-ray flash, Soderberg et al. 2005), GRB 060505, and GRB 060614 (Fynbo et al. 2006). Note that actually some "SGRBs" with so-called extended emission can also be classified as supernova-less long GRBs, but such events have also been included in previous studies of GW/SGRB association. Below we focus on the "traditional" long–short GRBs and introduce them in some detail.

XRF 040701 was localized by the Wide-Field X-Ray Monitor on board the High Energy Transient Explorer (HETE-2) on 2004 July 1.542 UT. It is characterized by the very low peak frequency (i.e., $\lt 6$ keV) of the prompt emission. The foreground extinction-corrected Hubble Space Telescope (HST) detection limit of Soderberg et al. (2005) is $\simeq 6$ mag fainter than SN 1998bw, the archetypal hypernova that accompanied long GRBs (Galama et al. 1998), at a redshift of z = 0.21. The analysis of the X-ray afterglow spectra reveals that the rest-frame extinction of the host galaxy is constrained to ${A}_{{\rm{V,host}}}\lt 2.8$ mag, suggesting that the associated supernova, if there was one, should be at least ~3.2 mag fainter than SN 1998bw (Soderberg et al. 2005). Due to the lack of sufficient multi-wavelength afterglow data, the "absence" of a bright supernova associated with XRF 040701 did not attract wide attention. The situation changed dramatically when the supernovae associated with GRB 060505 and GRB 060614 had not been detected down to limits hundreds of times fainter than SN 1998bw (Fynbo et al. 2006). In particular, GRB 060614, a bright burst with a duration of ~102 s at a redshift of 0.125, had dense follow-up observations with the Very Large Telescope and Hubble Space Telescope. The physical origin (either a peculiar collapsar or a compact object merger) of GRB 060614 was debated for years (e.g., Della Valle et al. 2006; Fynbo et al. 2006; Gal-Yam et al. 2006; Gehrels et al. 2006; Zhang et al. 2007). The re-analysis of the optical afterglow emission of GRB 060614 found significant excess components in multi-wavelength photometric observations, which can be reasonably interpreted as a Li–Paczyński macronova powered by the radioactive decay of debris following an NS–BH merger, while the weak supernova model does not work (Jin et al. 2015; Yang et al. 2015). As summarized in Xu et al. (2009), the origin of GRB 060505 at a redshift of z = 0.089 is less clear. The properties of its host galaxy seem to be consistent with those expected for long-duration GRBs (Thöne et al. 2008) but GRB 060505 is an outlier of the so-called Amati relation that holds for long GRBs (Amati et al. 2007). The HST observations at $t\sim 14.4$ days after the burst did not find optical emission down to limiting AB magnitudes of 27.3 in the F814W band and 27.1 in the F475W band (Ofek et al. 2007). Such stringent limits are strongly at odds with the collapsar model but can be quite consistent with the BNS merger model as long as the r-process ejecta has a mass $\lt {10}^{-3}\,{M}_{\odot }$ (Jin et al. 2016).

It may be still a bit early to conclude that all "nearby supernova-less" long GRBs (i.e., "long–short GRBs") are from mergers of compact object binaries. The successful identification of a macronova signal in the long–short event GRB 060614, nevertheless, renders such a possibility more attractive than before. If the NS–BH merger model for GRB 060614 is correct, the luminosity distance of this event is ${D}_{{\rm{L}}}\approx 576$ Mpc, which is smaller than ${D}_{* ,\mathrm{NS}\mbox{--}\mathrm{BH}}$ as long as ${\rho }_{* }\leqslant 10.8$ (see Equation (3)). For GRB 060505, the redshift z = 0.089 corresponds to a luminosity distance ${D}_{{\rm{L}}}\approx 400$ Mpc, which is almost equal to ${D}_{* ,\mathrm{BNS}}$ for ${\rho }_{* }=9$ (see Equation (1)). Intriguingly, among the supernova-less and short events detected so far (note that the GBM transient 150914 is still uncertain), the long–short bursts GRB 060505 and GRB 060614 are the only candidates that might yield a detectable GW signal for the advanced LIGO/VIRGO network (see Figure 1).

**Figure 1.** The "nearby" (i.e., $z\leqslant 0.3$ ) *supernova-less* GRBs, including the SGRBs and the long–short GRBs, and the averaged sensitivity distances of the advanced LIGO/VIRGO network (i.e., ${\rho }_{* }\geqslant 9$ ) for mergers of binary neutron stars and of a neutron star and stellar-mass black hole. The open circles are the SGRBs discussed in the literature, including GRB 050724 with a duration ${T}_{90}=3\pm 1\,{\rm{s}}$ (Fox et al. 2005). The red filled circles are the long–short GRBs. The data are taken from Berger (2014), Fynbo et al. (2006), Soderberg et al. (2005), and Levan et al. (2015). Interestingly, the long–short burst GRB 060505 is within ${D}_{* ,\mathrm{BNS}}({\rho }_{* }=9)$ . Three short bursts (GRB 061211, GRB 080905A, and GRB 150101B) and the long–short burst GRB 060614 are within ${D}_{* ,\mathrm{NS}\mbox{--}\mathrm{BH}}({\rho }_{* }=9)$ . Though the NS–BH merger rate is expected to be one order of magnitude lower than that of BNS mergers, and hence most supernova-less GRBs should originate from a BNS merger, there is some evidence for an NS–BH merger origin of GRB 060614. Hence the GW signals of GRB 060614-like events taking place in the era of advanced LIGO/VIRGO would be detectable. The adopted cosmological model parameters are presented in detail in Section 4.1. GW150914 is also included in the plot for illustration. We remind the readers that the observed gravitational wave event is a binary black hole coalescence. Although for O1 (first advanced LIGO observational run), advanced LIGO detectors are not yet sensitive enough to detect NS–BH events from that distance, this observation of GW150914 demonstrates that CBC events do happen within the NS–BH horizon of the full sensitive advanced LIGO/Virgo network.
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**Figure 1.** The "nearby" (i.e., $z\leqslant 0.3$ ) *supernova-less* GRBs, including the SGRBs and the long–short GRBs, and the averaged sensitivity distances of the advanced LIGO/VIRGO network (i.e., ${\rho }_{* }\geqslant 9$ ) for mergers of binary neutron stars and of a neutron star and stellar-mass black hole. The open circles are the SGRBs discussed in the literature, including GRB 050724 with a duration ${T}_{90}=3\pm 1\,{\rm{s}}$ (Fox et al. 2005). The red filled circles are the long–short GRBs. The data are taken from Berger (2014), Fynbo et al. (2006), Soderberg et al. (2005), and Levan et al. (2015). Interestingly, the long–short burst GRB 060505 is within ${D}_{* ,\mathrm{BNS}}({\rho }_{* }=9)$ . Three short bursts (GRB 061211, GRB 080905A, and GRB 150101B) and the long–short burst GRB 060614 are within ${D}_{* ,\mathrm{NS}\mbox{--}\mathrm{BH}}({\rho }_{* }=9)$ . Though the NS–BH merger rate is expected to be one order of magnitude lower than that of BNS mergers, and hence most supernova-less GRBs should originate from a BNS merger, there is some evidence for an NS–BH merger origin of GRB 060614. Hence the GW signals of GRB 060614-like events taking place in the era of advanced LIGO/VIRGO would be detectable. The adopted cosmological model parameters are presented in detail in Section 4.1. GW150914 is also included in the plot for illustration. We remind the readers that the observed gravitational wave event is a binary black hole coalescence. Although for O1 (first advanced LIGO observational run), advanced LIGO detectors are not yet sensitive enough to detect NS–BH events from that distance, this observation of GW150914 demonstrates that CBC events do happen within the NS–BH horizon of the full sensitive advanced LIGO/Virgo network.
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The presence of two Swift GRB candidates within the averaged sensitivity distance of the advanced LIGO/VIRGO network for ${\rho }_{* }\geqslant 9$ is indeed very encouraging for the ongoing GW experiments. On the other hand, if the supernova-less long-duration XRFs/GRBs were from a peculiar kind of collapsar (which is very unlikely to be the case for GRB 060614), we could verify it with the non-detection of a GW signal. Therefore we suggest that the supernova-less long-duration XRFs/GRBs are one of prime targets for the advanced LIGO/VIRGO network and the nature of such "mysterious" events would be unambiguously pinned down in the era of GW astronomy.

3. MODEL-DEPENDENT ESTIMATES OF TIME LAG BETWEEN BINARY COALESCENCE AND GRB ONSET

Though studied extensively, the launch, acceleration, and energy dissipation of the GRB ejecta are still heavily debated (see Zhang 2014, for a dedicated review). Instead of carrying out advanced study of a specific model, in this section we adopt some widely discussed scenarios and present our model-dependent estimates of the time lag between the GW coalescence time and GRB onset (i.e., ${\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB}}$ ) and examine how a statistical study of ${\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB}}$ could help us to better understand the physical processes taking place at the central engine.

In general, ${\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB}}$ can be divided into two parts. One is the time delay between the merger time, which could be estimated by analyzing the GW data (e.g., Fairhurst 2011; Veitch et al. 2015), and the successful launch of the ultrarelativistic ejecta (i.e., ${\rm{\Delta }}{t}_{{\rm{laun}}}$ ). The other is the time delay between the launch of the ultrarelativistic ejecta and the onset of gamma-ray emission (i.e., ${\rm{\Delta }}{t}_{{\rm{em}}}$ ). Below we examine ${\rm{\Delta }}{t}_{{\rm{laun}}}$ and ${\rm{\Delta }}{t}_{{\rm{em}}}$ separately under different models and then get the corresponding estimate of

$\begin{eqnarray*}&&{\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB}}={\rm{\Delta }}{t}_{{\rm{laun}}}+{\rm{\Delta }}{t}_{{\rm{em}}}.\end{eqnarray*}$

Notice that we are using the coalescence time, when the GW signal spikes, as a proxy for the moment of merger. One might reasonably argue that these two are not identical, but as the binary system evolves very rapidly toward the merger, they should not differ by more than a couple of rotations around the merger. For BNS systems the merger frequency is around 1000 Hz, so the difference in time cannot exceed ~1 ms (Fairhurst 2011; Pürrer 2014).

In this work we focus on the most widely adopted hypothesis—that the SGRBs were powered quickly after the merger (i.e., ${\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB}}\lt 10$ s). However, notice that a small number of SGRBs seem to have precursor emission, and the precursors are likely from the same central engine activity as the main bursts (Charisi et al. 2015). In the binary merger scenario, the merger may well have happened before the precursor. If so, the time lag between the GW signal and the SGRB/long–short GRB can be long, up to ~100 s, which may reflect the lifetime of the supramassive neutron star formed in the merger or alternatively the fall-back accretion timescale of the fragmented part of the compact object. Rezzolla & Kumar (2015) argued that the BNS mergers might actually take place several hours before the SGRBs. The observations of GW/GRB association can easily distinguish between such scenarios (i.e., ${\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB}}\sim {10}^{2}$ –10⁴ s) and the short-delay cases (i.e., ${\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB}}\lt 10$ s), as shown in Table 1. A long time lag between the GW and GRB signals, in principle, could also arise from the superluminal movement of the GW in the vacuum or its higher velocity than the photons in the gravitational potential. Such possibilities can be precisely tested as long as a sample of GW/GRB association is established (see Section 4.2 for the details).

Table 1. Expected Time Delay between Coalescence and GRB Onset (i.e., ${\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB}}$ )^a

Mergers	Prompt Remnant	${R}_{{\rm{pro}}}\ll {10}^{16}$ cm	${R}_{{\rm{pro}}}\sim {10}^{16}$ cm
BNS	BH	~10 ms	$\sim 2\,{\rm{s}}{(1+z)({R}_{{\rm{pro}}}/{10}^{16}{\rm{cm}})(\eta /300)}^{-2}$
	DRS^b HMNS	~100 ms	$\sim 0.1\,{\rm{s}}+2\,{\rm{s}}{(1+z)({R}_{{\rm{pro}}}/{10}^{16}{\rm{cm}})(\eta /300)}^{-2}$
	TPS^b HMNS	~1 s	$\sim 1\,{\rm{s}}+2\,{\rm{s}}{(1+z)({R}_{{\rm{pro}}}/{10}^{16}{\rm{cm}})(\eta /300)}^{-2}$
	GRB−DiffNS^c	$\sim 10$ –100 ms	$\sim 0.1\,{\rm{s}}+2\,{\rm{s}}{(1+z)({R}_{{\rm{pro}}}/{10}^{16}{\rm{cm}})(\eta /300)}^{-2}$

NS–BH	BH	~10 ms	$\sim 2\,{\rm{s}}{(1+z)({R}_{{\rm{pro}}}/{10}^{16}{\rm{cm}})(\eta /300)}^{-2}$

Notes.

^aNote that in some specific cases ${\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB}}\sim {10}^{2}$ ${\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB}}\sim {10}^{2}$ –10⁴ s is possible (e.g., Charisi et al. 2015; Rezzolla & Kumar 2015), which can be easily distinguished from the scenarios summarized in this table as long as a sample of GW/GRB association has been established. ^bDRS stands for "Differential rotation supported" and TPS for "Thermal pressure supported." ^cGRB−DiffNS represents the case in which a differentially rotating NS directly launches GRB outflow (see the last paragraph of Section 3.1.1 for the discussion of such a possibility). This case is different from the first three scenarios, in which the GRB ejecta is assumed to be launched when the gravitational collapse takes place.

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3.1. ${\rm{\Delta }}{t}_{{\rm{laun}}}$ Expected in Different Merger Scenarios

3.1.1. NS–NS Mergers

The maximum gravitational mass of a cold non-rotating neutron star is known to be ${M}_{{\rm{\max }}}\gt 2\,{M}_{\odot }$ (Antoniadis et al. 2013) and the threshold for collapse of the merger-formed remnants into BHs can be estimated roughly as ${M}_{{\rm{thres}}}\approx 1.35\,{M}_{{\rm{\max }}}$ (Shibata & Taniguchi 2006). Hence a total gravitational mass $\gtrsim 2.7\,{M}_{\odot }$ is likely required for prompt collapse to a BH. Such massive neutron star binaries should account for just a small fraction of merger events if the mass distribution of "cosmic" neutron star binaries resembles what has been observed in the Galaxy (see Lattimer 2012, for a recent review). Hence usually we do not expect the prompt collapse of the merger-formed neutron stars. Instead the merger-formed remnants are widely expected to be very massive neutron stars with strong differential rotation that can support themselves against collapse at least temporarily.

Such remnants are called hypermassive neutron stars (HMNSs). The fate of post-merger HMNSs, however, is uncertain and is contingent on the mass limit for support of a hot, differentially rotating configuration (e.g., Baumgarte et al. 2000; Hotokezaka et al. 2013b). Below we present model-dependent estimates of the collapse time of HMNSs, which we regard as ${\rm{\Delta }}{t}_{{\rm{laun}}}$ . One exception will be presented in the last paragraph of this subsubsection.

When ${M}_{{\rm{\max }}}\lt M\lt {M}_{{\rm{thres}}}$ , various mechanisms can act to dissipate and/or transport energy and angular momentum, possibly inducing collapse after a delay that could range from tens of milliseconds to a few seconds (see Faber & Rasio 2012, for a recent review). For instance, in the presence of a strong magnetic field, the magnetic braking effect can effectively transfer the angular momentum in a timescale ${\tau }_{{\rm{br}}}\sim {R}_{{\rm{s}}}{/V}_{{\rm{A}}}\sim \,0.3\,{\rm{s}}$ $({R}_{{\rm{s}}}{/10}^{6}\,{\rm{cm}})$ ${(\rho {/10}^{15}{\rm{g}}{\mathrm{cm}}^{-3})}^{\displaystyle \frac{1}{2}}$ $(\epsilon /0.3)$ ${({B}_{{\rm{s}}}{/10}^{13}{\rm{G}})}^{-1}$ , where ${V}_{{\rm{A}}}$ is the Alfvén velocity, ${R}_{{\rm{s}}}$ is the radius of the neutron star, and $\epsilon \sim 0.3$ is the expected strength ratio between the surface magnetic field ${B}_{{\rm{s}}}$ and the interior poloidal magnetic field (Shapiro 2000). Another mechanism is magnetorotational instability (MRI), which generates turbulence in a magnetized rotating fluid body that amplifies the magnetic field and transfers angular momentum. In the presence of MRI, an effective viscosity is likely to be generated with the effective viscous parameter ${\nu }_{{\rm{vis}}}\sim {\alpha }_{{\rm{vis}}}{c}_{{\rm{s}}}^{2}/{{\rm{\Omega }}}_{{\rm{c}}}$ , where ${\alpha }_{{\rm{vis}}}$ is the viscosity parameter, ${c}_{{\rm{s}}}$ is the sound velocity of the envelope of the HMNS, and ${{\rm{\Omega }}}_{{\rm{c}}}$ is the angular velocity of the core of the differentially rotating neutron star (Balbus & Hawley 1991). Thus, the timescale for viscous angular momentum transport can be estimated as ${\tau }_{{}_{{\rm{MRI}}}}\sim {R}_{{\rm{s}}}^{2}{/\nu }_{{\rm{vis}}}\sim 0.1\,{\rm{s}}$ ${({R}_{{\rm{s}}}/{10}^{6}{\rm{cm}})}^{2}$ ${({\alpha }_{{\rm{vis}}}/0.01)}^{-1}$ ${({c}_{{\rm{s}}}/0.1c)}^{-2}{({{\rm{\Omega }}}_{{\rm{c}}}/{10}^{4}\mathrm{rad}{{\rm{s}}}^{-1})}^{-1}$ (Hotokezaka et al. 2013b; Siegel et al. 2013). A reasonable estimate of the termination timescale of the differential rotation is ${\tau }_{{\rm{diff}}}=\min \{{\tau }_{{\rm{br}}},{\tau }_{{}_{{\rm{MRI}}}}\}\sim 0.1\,{\rm{s}}$ , after which the HMNS is expected to collapse.

The situation is even less uncertain when both finite-temperature effects in the equation of state and neutrino emission of the central compact object have been taken into account. In the numerical simulations of the merger of binary neutron stars performed in full general relativity incorporating the finite-temperature effect and neutrino cooling, Sekiguchi et al. (2011) found that the effect of the thermal energy is significant and can increase Mmax by a factor of 20%–30% for a high-temperature state with T ≥ 20 MeV. Since they are not supported by differential rotation, the hypermassive remnants were predicted to be stable until neutrino cooling, with luminosity of ~3–10 × 10⁵³ erg s⁻¹ has removed the pressure support in τ_thermal ~1 s (Sekiguchi et al. 2011).

For $t\lt {\tau }_{{\rm{w}}}$ , a baryon-loaded wind is continuously ejected, which bounds the bulk Lorentz factor of the jet to ${{\rm{\Gamma }}}_{{\rm{w}}}\sim 5({L}_{{\rm{jet}}}{/10}^{52}\,\mathrm{erg}\,{{\rm{s}}}^{-1}){({\dot{M}}_{{\rm{w}}}{/10}^{-3}{M}_{\odot }{{\rm{s}}}^{-1})}^{-1}$ , where ${L}_{{\rm{jet}}}$ is the isotropic-equivalent luminosity of the jet, ${\dot{M}}_{{\rm{w}}}$ is the mass loss rate via the wind, and ${\tau }_{{\rm{w}}}$ is the wind duration (either ${\tau }_{{\rm{diff}}}$ or ${\tau }_{{\rm{thermal}}}$ , depending on the mechanism that mainly supports the star against collapse). Such a low ${{\rm{\Gamma }}}_{{\rm{w}}}$ is too small to give rise to energetic GRB emission. Hence it is widely anticipated that no GRB is possible unless the neutrino-driven wind gets very weak or more realistically the neutron star has collapsed to a BH. After the collapse of the HMNS, the earlier outward-moving dense wind remains to hamper the advance of the jet, whose injection lifetime, ${t}_{{\rm{jet}}}$ , is determined by the viscous timescale of the accretion disk. Murguia-Berthier et al. (2014) suggested that in the BH central engine model for (short) GRBs, the BH formation should occur promptly because any moderate delay at the stage of a hypermassive neutron star would result in a choked jet. The argument that just the mergers with a remnant collapse within a timescale $\sim {t}_{{\rm{diff}}}\sim 0.1\,{\rm{s}}$ can produce (short) GRBs may hold only in the scenario of energy extraction via neutrino mechanisms. The accretion timescale of the torus formed in binary neutron star mergers can be estimated as ${t}_{{\rm{acc}}}\sim 0.1\,{\rm{s}}\,{({\alpha }_{{\rm{vis}}}/0.1)}^{-6/5}$ (Popham et al. 1999; Narayan et al. 2001). Following Zalamea & Beloborodov (2011) and Fan & Wei (2011), it is straightforward to estimate the corresponding luminosity of the annihilated neutrinos/antineutrinos as ${L}_{\nu \bar{\nu }}\approx {10}^{49}\,\mathrm{erg}\,{{\rm{s}}}^{-1}{(\dot{m}/0.1{M}_{\odot }{{\rm{s}}}^{-1})}^{9/4}$ , where $\dot{m}$ is the accretion rate and the spin of the BH has been taken to be a = 0.78, a typical value for BHs formed in binary neutron star mergers. The isotropic-equivalent luminosity of the ejecta is then ${L}_{{\rm{jet}}}\approx 2\times {10}^{51}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ ${(\dot{m}/0.1{M}_{\odot }{{\rm{s}}}^{-1})}^{9/4}$ ${({\theta }_{{\rm{jet}}}/0.1)}^{-2}$ , which satisfies the condition of relativistic expansion of the jet head within the preceding neutrino-driven wind medium (i.e., Equation (8) of Murguia-Berthier et al. 2014) as long as $\dot{m}\gt {\dot{m}}_{{\rm{jet}}}\approx 0.2\,{M}_{\odot }\,{{\rm{s}}}^{-1}$ . The mass of the accretion disk is ${M}_{{\rm{disk}}}\sim {\dot{m}}_{{\rm{jet}}}{t}_{{\rm{acc}}}\sim 0.02\,{M}_{\odot }$ . Such a mass is consistent with that found in numerical simulations of binary neutron star mergers (Faber & Rasio 2012; Nagakura et al. 2014), which in turn suggests that short GRBs are possible for ${t}_{{\rm{jet}}}\approx {t}_{{\rm{acc}}}\gt {\tau }_{{\rm{w}}}$ if ${\tau }_{{\rm{w}}}\lt 0.1\,{\rm{s}}$ , in agreement with Murguia-Berthier et al. (2014). If instead ${\tau }_{{\rm{w}}}\gg 0.1\,{\rm{s}}$ , the required ${M}_{{\rm{disk}}}$ would be too massive to be realistic (Fan & Wei 2011; Liu et al. 2015).

The situation is significantly different for the magnetic process to launch the GRB ejecta. The huge amount of rotational energy of the BH can be extracted efficiently via the Blandford−Znajek process and the luminosity of the electromagnetic outflow can be estimated from ${L}_{{\rm{BZ}}}\approx 6\times {10}^{49}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\,{(a/0.75)}^{2}{({B}_{{\rm{H}}}/{10}^{15}{\rm{G}})}^{2}$ , where ${B}_{{\rm{H}}}\sim 1.1\times {10}^{15}{\rm{G}}\,{(\dot{m}/0.01{M}_{\odot }{{\rm{s}}}^{-1})}^{1/2}{({R}_{{\rm{H}}}/{10}^{6}{\rm{cm}})}^{-1}$ is the magnetic field strength on the horizon of the BH (Blandford & Znajek 1977). Therefore $\dot{m}\sim 0.01\,{M}_{\odot }\,{{\rm{s}}}^{-1}$ is sufficient to launch energetic ejecta with ${L}_{{\rm{jet}}}\sim {10}^{52}\,\mathrm{erg}\,{{\rm{s}}}^{-1}{({\theta }_{{\rm{j}}}/0.1)}^{-2}$ . An $\alpha \leqslant 0.01$ is needed to get a ${t}_{{\rm{acc}}}\sim$ a few seconds. Such a "small" α is still possible (Narayan et al. 2001) and the required mass of the accretion disk is also in the reasonable region of $\sim 0.01\,{M}_{\odot }$ . Note that in these estimates the ejecta "breakout" criterion suggested in Murguia-Berthier et al. (2014) has been adopted. In reality, the Poynting-flux jet could break out from the "neutrino-driven wind" more easily than the hydrodynamic jet. This is because the reverse shock that slows down the hydrodynamic jet and the collimation shock that collimates it cannot form within the Poynting-flux-dominated jet. As a result the Poynting-flux-dominated jet moves much faster and dissipates much less energy while it crosses the preceding neutrino-driven wind (Bromberg et al. 2014). The latest time-dependent 3D relativistic magnetohydrodynamic simulations of relativistic, Poynting-flux-dominated jets that propagate into a medium with a spherically symmetric power-law density distribution have shown that some instabilities can lead to efficient dissipation of the toroidal magnetic field component, and hence the propagation of such a "headed" magnetized ejecta is likely similar to that of a hydrodynamic ejecta (Bromberg & Tchekhovskoy 2015). In such a case, the "breakout" criterion of Murguia-Berthier et al. (2014) applies. After the "breakout" of the "headed" magnetized ejecta, an evacuated funnel is formed and the later ejecta moves freely without significant magnetic energy dissipation (i.e., it is within the phase of the "headless" jet (Bromberg & Tchekhovskoy 2015)). Therefore, a ${t}_{{\rm{acc}}}\sim$ a few seconds may be sufficiently long to successfully produce GRBs for ${\tau }_{{\rm{w}}}\sim {\tau }_{{\rm{thermal}}}\sim 1\,{\rm{s}}$ . The conclusion of this paragraph is that a GRB is still possible in the case of ${\tau }_{{\rm{w}}}\sim {\tau }_{{\rm{thermal}}}\sim 1\,{\rm{s}}$ but the outflow should be launched via magnetic processes.

The expected time delay between the merger of the binary neutron stars and the launch of the ultrarelativistic GRB outflow can thus be approximately summarized as ${\rm{\Delta }}{t}_{{\rm{laun,BNS}}}\sim (0.01,\,0.1,\,1\,{\rm{s}})$ for the prompt formation of a BH, a HMNS supported by differential rotation, and a HMNS supported by thermal pressure, respectively. The minimum ${\rm{\Delta }}{t}_{{\rm{laun,BNS}}}$ is taken to be ~10 ms since the merger time is expected to be measured with an accuracy better than ~10 ms and the ultrarelativistic outflow may be launched promptly.

In the above discussion we assume that the short GRBs are produced when the HMNSs collapse into BHs. There is another possibility—that the differentially rotating NSs can eject significant material toward the rotation axis, which might also produce (short) GRBs. Such a scenario has attracted wide attention since the analysis of a good fraction of afterglow emission from Swift short GRBs found possible evidence for the magnetar central engine (Rowlinson et al. 2013). One possible physical scenario is that the differentially rotating neutron star wraps the poloidal seed magnetic field into superstrong toroidal fields ( ${B}_{{\rm{f}}}\sim {10}^{17}$ G), which may emerge from the star through buoyancy and then generate GRBs via magnetic energy dissipation (Kluzániak & Ruderman 1998; Dai et al. 2006). In this model ${\rm{\Delta }}{t}_{{\rm{laun,BNS}}}$ is expected to be the time needed to amplify the seed magnetic field to ${B}_{{\rm{f}}}\sim {10}^{17}$ G, i.e., ${\rm{\Delta }}{t}_{{\rm{laun,BNS}}}\sim 5\,{\rm{ms}}$ $({B}_{{\rm{f}}}/{10}^{17}\,{\rm{G}})$ $(\epsilon /0.3)$ ${({B}_{{\rm{s}}}/{10}^{15}{\rm{G}})}^{-1}$ ${({\rm{\Delta }}{\rm{\Omega }}/6000\mathrm{rad}{{\rm{s}}}^{-1})}^{-1}$ , where ${\rm{\Delta }}{\rm{\Omega }}\equiv 2\pi (1/{P}_{{\rm{c}}}-1/{P}_{{\rm{s}}})$ , and ${P}_{{\rm{c}}}$ and ${P}_{{\rm{s}}}$ are the rotational periods of the differentially rotating internal part and the main NS, respectively (Kluzániak & Ruderman 1998). With the ${B}_{{\rm{s}}}$ and the initial rotational periods of the magnetar central engine estimated in Rowlinson et al. (2013) we have ${\rm{\Delta }}{t}_{{\rm{laun,BNS}}}\sim 10$ –100 ms.

3.1.2. NS–BH Mergers

In this case the central engine is a stellar-mass BH, and the region along the spin axis of the BH is likely cleaner than in the case of NS–NS mergers. However, the joint effects of shocks during the disk circularization, instabilities at the disk/tail interface, and neutrino absorption unbind a small amount ( $\sim {10}^{-4}\,{M}_{\odot }$ ) of material in the polar regions (Foucart et al. 2015). Over longer timescales, the neutrino-powered winds become active and eject material in the polar regions. Though the material is still negligible compared to the material ejected dynamically in the equatorial plane during the disruption of the neutron star, this ejecta could impact the formation of a relativistic jet (Foucart et al. 2015). Nevertheless, a few per cent of the energy radiated in neutrinos is expected to be deposited in the region along the spin axis of the BH through $\nu \bar{\nu }$ annihilations (Janka et al. 1999; Setiawan et al. 2006). Energy deposition at a rate $\sim {10}^{51}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ might also be able to power a short γ-ray burst (Lee & Ramirez-Ruiz 2007; Foucart et al. 2014).

As in the BNS merger scenario, the magnetic mechanism may be more promising in launching ultrarelativistic outflows and then giving rise to GRBs. In the recent high-resolution numerical-relativity simulations for the merger of BH–NS binaries that are subject to tidal disruption and subsequent formation of a massive accretion torus, the accretion torus formed quickly and the magnetic field was amplified significantly due to the non-axisymmetric MRI and magnetic winding (Kiuchi et al. 2015; Paschalidis et al. 2015). The amplification can yield $B\sim {10}^{15}$ G at the BH poles in ~20 ms after the merger, and the corresponding Blandford–Znajek luminosity can be sufficiently high to power GRBs.

The data from GRB 060614 have likely shed valuable light on the role of the magnetic process in extracting the energy for the GRBs. Such a long–short event is most likely powered by the merger of a binary system of a neutron star and stellar-mass BH (Jin et al. 2015; Yang et al. 2015). As found in various numerical simulations, the total mass of the accretion disk is expected to be not much more than $\sim 0.1\,{M}_{\odot }$ . On the other hand, the duration of the "long-lasting" soft γ-ray emission is ~100 s. Hence the time-averaged accretion rate is expected to be just of the order of $\dot{m}\sim {10}^{-3}\,{M}_{\odot }\,{{\rm{s}}}^{-1}$ . For such a low accretion rate, the neutrino mechanism is expected to be unable to launch energetic GRB outflow (e.g., Fan et al. 2005; Liu et al. 2015). Instead, the Blandford−Znajek process can give rise to Poynting-flux-dominated outflow with an "intrinsic" luminosity ${L}_{{\rm{BZ}}}\approx 6\times {10}^{47}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ ${(a/0.75)}^{2}$ ${({B}_{{\rm{H}}}/{10}^{14}{\rm{G}})}^{2}$ (Blandford & Znajek 1977), which is sufficient to explain the observed γ-ray luminosity of GRB 060614 after the correction of the jet opening angle of the outflow ${\theta }_{{\rm{j}}}\sim 0.1$ (see Xu et al. 2009). Therefore, the soft long-lasting gamma-ray "tail" emission of GRB 060614 likely has a moderate to high linear polarization (Fan et al. 2005).

In view of these facts, we suggest that ultrarelativistic outflows may be launched within ${\rm{\Delta }}{t}_{{\rm{laun,BHNS}}}\sim 10$ ms after BH–NS mergers via either neutrino–antineutrino annihilation or magnetic process(es).

3.2. ${\rm{\Delta }}{t}_{{\rm{em}}}$ Expected in Models of Baryonic and Magnetic Outflow

3.2.1. The Bayronic Outflow

The neutrino–antineutrino annihilation process will launch an extremely hot fireball. For a baryonic outflow of this kind, the acceleration is well understood (Mészáros et al. 1993; Piran et al. 1993) and most of the initial thermal energy may have been converted into kinetic energy of the baryons at the end of the acceleration (Shemi & Piran 1990). A quasithermal emission component, however, is likely inevitable (see Chhotray & Lazzati 2015, and the references therein for the resulting spectrum). The quasithermal emission is mainly from the photosphere at a radius ${R}_{{\rm{ph}}}$ , which can be estimated as ${R}_{{\rm{ph}}}\approx 4.6\times {10}^{10}\,{\rm{cm}}\,{(L/{10}^{51}\mathrm{erg}{{\rm{s}}}^{-1})(\eta /200)}^{-3}$ , where L is the total isotropic-equivalent luminosity of the baryonic outflow and $\eta \sim {10}^{2}$ –10³ is the initial dimensionless entropy (Paczyński 1990; Daigne & Mochkovitch 2002). Assuming an initial launch radius of the fireball ${R}_{0}\approx {10}^{7}$ cm, at ${R}_{{\rm{ph}}}$ the thermal radiation luminosity is expected to be ${L}_{{\rm{th}}}\approx 2.5\times {10}^{49}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ ${(L/{10}^{51}\mathrm{erg}{{\rm{s}}}^{-1})}^{1/3}$ ${(\eta /200)}^{8/3}$ ${({R}_{0}/{10}^{7}{\rm{cm}})}^{2/3}$ . And the quasithermal radiation peaks at a temperature ${T}_{{\rm{th}}}\sim 80\,{\rm{keV}}$ ${(L/{10}^{51}\mathrm{erg}{{\rm{s}}}^{-1})}^{1/4}$ ${(\eta /200)}^{2/3}{({R}_{0}/{10}^{7}{\rm{cm}})}^{1/6}$ . For the sources within the detection ranges of advanced LIGO/VIRGO (i.e., $D\approx 300$ Mpc), the energy flux can be as high as ${ \mathcal F }\sim 2\times {10}^{-6}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ ${(L/{10}^{51}\mathrm{erg}{{\rm{s}}}^{-1})}^{1/3}$ ${(\eta /200)}^{8/3}$ ${({R}_{0}/{10}^{7}{\rm{cm}})}^{2/3}$ , which is detectable by Swift or the Fermi GBM. The acceleration timescale of the baryonic outflow is $\sim (1+z){R}_{0}/c$ and the delay between the termination of the acceleration and the emergence of the thermal photons can be estimated as $\sim (1+z){R}_{{\rm{ph}}}/2{\eta }^{2}c$ . For $\eta \geqslant 100$ the latter is significantly smaller than the former, hence ${\rm{\Delta }}{t}_{{\rm{em}}}\sim (1+z){R}_{0}/c\sim 0.3\,{\rm{ms}}\,(1+z)({R}_{0}/{10}^{7}\,{\rm{cm}})$ , which is negligibly small.

If the photospheric quasithermal radiation is undetectable (say, for NS–BH mergers at $D\sim 1$ Gpc), more efficient emission may be cased by collisions between the baryonic shells ejected from the same central engine but with very different Lorentz factors. Strong internal shocks are generated and ultrarelativistic particles are accelerated. A fraction of the internal shock energy has been converted into a magnetic field, and electrons moving in the magnetic field produce energetic γ-ray emission (Rees & Mészáros 1994). In such a model, the variability of the prompt emission largely traces the behavior of the activity of the central engine, and the onset of the "internal shock emission" is expected to be within the typical variability timescale of the prompt emission, which can be ~1–10 ms (Piran 1999), i.e., ${\rm{\Delta }}{t}_{{\rm{em}}}\sim 1$ –10 ms.

3.2.2. The Magnetic Outflow

In the case of the magnetic outflow, both the acceleration and the subsequent dissipation/radiation of energy are more uncertain (see Granot et al. 2015; Kumar & Zhang 2015, for recent reviews). If the magnetic energy has been effectively converted into kinetic energy of the outflow (Granot et al. 2005), the prompt emission of GRBs can be from magnetized internal shocks (Fan et al. 2004) or from the photosphere with internal dissipation of energy via gradual magnetic reconnection (Giannios 2008), and we expect ${\rm{\Delta }}{t}_{{\rm{em}}}\sim 1$ –10 ms. Note that the latest time-dependent 3D relativistic magnetohydrodynamic simulations of relativistic, Poynting-flux-dominated jets that propagate into a medium with a spherically symmetric power-law density distribution have found that some instabilities can lead to efficient dissipation of the toroidal magnetic field component (Bromberg & Tchekhovskoy 2015), for which the onset of prompt emission is likely dominated by photospheric radiation and ${\rm{\Delta }}{t}_{{\rm{em}}}$ is negligibly small.

If the photospheric radiation is too weak to be detectable by Fermi GBM-like detectors, due to either the absence of a dense wind-like medium in the direction of the BH spin or the small luminosity of the material that is breaking out, the observed onset of the prompt emission is likely significantly delayed. In some models most of the initial magnetic energy has not been converted into the kinetic/thermal energy of the outflow, and the prompt emission of GRBs is due to the dissipation of magnetic energy at a rather large distance ${R}_{{\rm{pro}}}\sim {10}^{16}$ cm possibly⁵ because of the breakdown of the magnetohydrodynamic approximation of the highly magnetized outflow (Usov 1994; Zhang & Mészáros 2002; Fan et al. 2005), or the current-driven instabilities developed in the outflow shell (Lyutikov & Blandford 2003), or the internal collision-induced magnetic reconnection and turbulence (Zhang & Yan 2011). Correspondingly we have ${\rm{\Delta }}{t}_{{\rm{em}}}\approx (1+z)$ ${R}_{{\rm{pro}}}/2{\eta }^{2}c\,\approx 2\,{\rm{s}}(1+z)$ ${({R}_{{\rm{pro}}}/{10}^{16}{\rm{cm}})(\eta /300)}^{-2}$ (i.e., the radial timescale), which is equal to the "angular timescale" of the emission, the minimal timescale of the GRB duration, as long as the ejecta has an opening angle larger than $1/\eta$ (Piran 1999). Since the angular timescale is derived for an infinitely thin radiating shell, it is expected to be shorter than the duration of the prompt emission of the whole GRB (i.e., ${\rm{\Delta }}{t}_{{\rm{em}}}\leqslant {T}_{90}$ ).

SGRB 050509B and SGRB 050709 have ${T}_{90}=0.04\,{\rm{s}}$ and 0.07 s, respectively (Fox et al. 2005). For such "brief" events, ${R}_{{\rm{pro}}}\sim {10}^{16}$ cm is disfavored unless $\eta \gtrsim 2000$ . A value of η as high as ~2000, however, would render them the outstanding outliers of the correlation $\eta \approx 250\,{({L}_{\gamma }/{10}^{52}\mathrm{erg}{{\rm{s}}}^{-1})}^{0.3}$ that holds for some long GRBs (Fan et al. 2012; Lü et al. 2012; Liang et al. 2015) and possibly also the short GRB 090510 if it has $\eta \gtrsim 1200$ , as argued in Ackermann et al. (2010), where L_γ is the γ-ray luminosity of the GRB. Moreover, unless there is some fine tuning whereby the central engine shuts down at almost the same time as the outflow breaks out of the dense material, the central engines of these two very short bursts should be (promptly formed) BHs and ${\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB}}\lt {T}_{90}$ is expected. For SGRB 050709 the modeling of the macronova signal favors an NS–BH merger origin (Jin et al. 2016), which supports our current argument (i.e., the central engine of SGRB 050709 was a BH).

3.3. Expected Relationship between ${\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB}}$ and T₉₀

We summarize in Table 1 the suspected ${\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB}}$ (i.e., the sum of ${\rm{\Delta }}{t}_{{\rm{laun}}}$ and ${\rm{\Delta }}{t}_{{\rm{em}}}$ ), where the case of ${R}_{{\rm{pro}}}\ll {10}^{16}$ cm includes the scenarios of photospheric radiation and regular (magnetized) internal shock radiation. Clearly, the shortest delay is expected in the cases of prompt BH formation in the NS–NS mergers or the NS–BH mergers if the onset of the prompt emission is governed by the photosphere or regular internal shocks (i.e., ${R}_{{\rm{pro}}}\ll {10}^{16}$ cm), and such events will be valuable in imposing very stringent constraints on the difference between the GW and the speed of light (see Section 4 for the details).

In Section 3.1.1 we have already mentioned that the prompt formation of BH in BNS mergers is likely uncommon. As long as ${R}_{{\rm{pro}}}\ll {10}^{16}$ cm, one naturally expects that (1) for BNS mergers ${\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB}}$ is significantly longer than that of the NS–BH mergers, i.e., ${\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB}}(\mathrm{NS}\mbox{--}\mathrm{BH})\ll {\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB}}({\rm{BNS}});$ (2) for NS–BH mergers, usually ${\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB}}$ is expected to be shorter than T₉₀, i.e., ${\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB}}(\mathrm{NS}\mbox{--}\mathrm{BH})\lt {T}_{90}$ . For ${R}_{{\rm{pro}}}\sim {10}^{16}\,{\rm{cm}}$ , we expect that ${\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB}}$ should be comparable with T₉₀ for both BNS and NS–BH merger-powered SGRBs. (Note that for some very brief events such as SGRB 050509B and SGRB 050709, ${R}_{{\rm{pro}}}\sim {10}^{16}$ cm is most likely disfavored.) Therefore, with reasonably large GRB samples from BNS mergers and from NS–BH mergers, the statistical distribution of ${\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB}}$ and T₉₀ for each sample, or alternatively the distribution of ${\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB}}$ for the combined sample, could shed valuable light on the physics of the central engine.

Are BNS mergers and NS–BH merger events distinguishable in the era of advanced LIGO/VIRGO? It is known that GW observations can efficiently measure the binary's chirp mass ${\boldsymbol{ \mathcal M }}\equiv {({m}_{1}{m}_{2})}^{3/5}/{({m}_{1}+{m}_{2})}^{1/5}$ , where m₁ and m₂ are the gravitational masses of the binary stars, although this leaves the individual masses undetermined (e.g., Bartos et al. 2013; Hannam et al. 2013). Moreover, the accuracy of the reconstruction of the masses is decreased by the additional degeneracy between mass ratio and spin. Fortunately, in many cases the nature of the binary system can be determined. For instance, if we consider non-spinning compact objects and ${\rho }_{* }\approx 10$ , then ${\boldsymbol{ \mathcal M }}\gtrsim 2.8{M}_{\odot }$ implies that one of the binary compact objects has to have a mass $\gt 3.2\,{M}_{\odot }$ , above ${M}_{{\rm{\max }}}$ for any reasonable NS models, while ${\boldsymbol{ \mathcal M }}\lesssim 1.2{M}_{\odot }$ suggests that the mass of each compact object needs to be $\lt 2{M}_{\odot }$ unless one of the NSs is smaller than $1{M}_{\odot }$ , in which case the limit on the heavier object is $3{M}_{\odot }$ (Bartos et al. 2013; Hannam et al. 2013). Together with the expected detection rate of GRBs with GW signals (see Section 2.1), we think that in the era of GW astronomy, reasonably large GRB samples from BNS mergers and from NS–BH mergers will be available and our goals will be (at least partly) achievable.

4. MEASURING THE VELOCITY OF THE GRAVITATIONAL WAVE AND TESTING EINSTEIN'S EQUIVALENCE PRINCIPLE (EEP)

4.1. Measuring the GW Velocity: the "Canonical" Approach

According to general relativity, in the limit in which the wavelength of gravitational waves is small compared to the radius of curvature of the background spacetime, the waves propagate with the velocity of light, i.e., c (see Will 1998, and references therein). In other theories, the speed ${v}_{{\rm{g}}}$ could differ from c. Let us define the parameter

$\begin{eqnarray}&&\varsigma \equiv (c-{v}_{{\rm{g}}})/c.\end{eqnarray} \tag{ 4 }$

If the gravitational wave velocity is subluminal (i.e., $\varsigma \gt 0$ ), then cosmic rays lose their energy via gravitational Cerenkov radiation significantly. The detection of ultrahigh-energy cosmic rays thus imposes a stringent constraint $0\leqslant \varsigma \leqslant 2\times {10}^{-15}\,(\mathrm{or}\,\mathrm{even}\,2\times \ {10}^{-19})$ (i.e., the "subluminal constraint"), depending on the Galactic or extragalactic origin of such particles (Caves 1980; Moore & Nelson 2001). However, there is no theoretical argument (or pathology) against GWs propagating faster than light (see Nishizawa & Nakamura 2014; Blas et al. 2016, and references therein) and the weak bounds from radiation damping in binary systems are $\varsigma \gt -0.01$ (Yagi et al. 2014). The time lag of arrival times between the GW and the simultaneously radiated photons is

$\begin{eqnarray}\delta {t}_{{\rm{o}}} & = & \displaystyle \frac{1}{c}{\displaystyle \int }_{0}^{{z}_{{\rm{o}}}}(1+z)\left(\displaystyle \frac{c}{{v}_{{\rm{g}}}}-1\right){dl}\\ & = & \displaystyle \frac{1}{{H}_{0}}{\displaystyle \int }_{0}^{{z}_{{\rm{o}}}}\left(\displaystyle \frac{\varsigma }{1-\varsigma }\right)\displaystyle \frac{{dt}}{\sqrt{{{\rm{\Omega }}}_{{\rm{M}}}{(1+z)}^{3}+{{\rm{\Omega }}}_{{\rm{\Lambda }}}}},\end{eqnarray} \tag{ 5 }$

where ${dl}={cdz}/[(1+z){H}_{0}\sqrt{{{\rm{\Omega }}}_{{\rm{M}}}{(1+z)}^{3}+{{\rm{\Omega }}}_{{\rm{\Lambda }}}}]$ is the differential distance the photons have traveled. Note that in this work we take the flat cosmological model (i.e., ${{\rm{\Omega }}}_{{\rm{M}}}+{{\rm{\Omega }}}_{{\rm{\Lambda }}}=1$ ), ${{\rm{\Omega }}}_{{\rm{M}}}=0.315$ , and ${H}_{0}\approx 70\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1}$ is Hubble's constant (Riess et al. 2011; Ade et al. 2014). In general, may be a function of the GW frequency (f) and especially when graviton mass is non-zero (i.e., ${m}_{{\rm{g}}}\gt 0$ ), which gives $\varsigma \approx {m}_{{\rm{g}}}^{2}{c}^{4}/2{h}^{2}{f}^{2}$ , where h is Planck's constant. For simplicity we assume a constant and focus on the association of GWs with electromagnetic counterparts at redshifts $z\ll 1$ . Hence Equation (5) yields (see also Will 1998; Nishizawa & Nakamura 2014)

$\begin{eqnarray}&&\varsigma \approx 5\,\times \,{10}^{-17}\,\left(\displaystyle \frac{200\,{\rm{Mpc}}}{D}\right)\left(\displaystyle \frac{\delta {t}_{{\rm{o}}}}{1\,{\rm{s}}}\right).\end{eqnarray} \tag{ 6 }$

In reality the photons and the coalescence are not usually simultaneous and we have $\delta {t}_{{\rm{o}}}=(1+z){\rm{\Delta }}{t}_{{\rm{e}}}-{\rm{\Delta }}{t}_{\mathrm{GW}-\mathrm{ph}}$ , where ${\rm{\Delta }}{t}_{\mathrm{GW}-\mathrm{ph}}$ and ${\rm{\Delta }}{t}_{{\rm{e}}}$ are the differences in arrival time and emission time, respectively, of the GW and the photons. In most cases, it is rather hard to get an a priori value for ${\rm{\Delta }}{t}_{{\rm{e}}}$ . Assuming ${\rm{\Delta }}{t}_{{\rm{e}}}=0$ (i.e., the GW and the electromagnetic counterparts were emitted simultaneously; see Nishizawa (2016) for a more general discussion), we constrain the absolute amplitude of as

$\begin{eqnarray}&&| \varsigma | \lt 5\,\times \,{10}^{-17}\,\left(\displaystyle \frac{200\,{\rm{Mpc}}}{D}\right)\left(\displaystyle \frac{{\rm{\Delta }}{t}_{\mathrm{GW}-\mathrm{ph}}}{1\,{\rm{s}}}\right).\end{eqnarray} \tag{ 7 }$

We call the above process the "canonical approach" to measuring the GW velocity directly, in which the graviton and photon are assumed to make the same journey (i.e., EEP is guaranteed). The advantage is that as long as a GW/GRB association is established one can constrain $| \varsigma |$ directly. In Section 4.2 we outline an approach to measure the GW velocity with a simultaneous test of EEP.

Some widely discussed electromagnetic counterparts of compact object mergers include (Metzger & Berger 2012): (a) (short) gamma-ray bursts and X-ray flares; (b) afterglow emission of the (off-beam) gamma-ray burst outflows; (c) macronova/kilonova emission of the sub-relativistic r-process material ejected during the merger; (d) radio radiation of the forward shock driven by the sub-relativistic outflow launched during the merger. These scenarios hold for both NS–NS and NS–BH mergers (note that for systems with very massive BHs, the NSs would be swallowed entirely and no bright electromagnetic counterparts would be expected). To constrain $| \varsigma |$ (see Equation (7)), the time delay between the merger and the "emergence" of the electromagnetic counterpart (i.e., ${\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB}}$ ) is needed.

The intense gamma-ray emission is expected to occur within seconds after the merger (see Table 1 for the model-dependent estimate, where ${\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB}}$ is the same as ${\rm{\Delta }}{t}_{\mathrm{GW}-\mathrm{ph}}$ needed in Equation (7)). The X-ray flares may appear within tens of seconds and sometimes may last $\sim {10}^{3}$ s or even longer. The challenge in detecting the "orphan" soft X-ray signal is the lack of X-ray detector(s) with a wide field of view pending the successful performance of Einstein Probe (http://ep.bao.ac.cn;Yuan et al. 2014) after 2022. Since advanced LIGO/VIRGO are expected to run fully in 2019, here we just focus on the detectors that may (still) work at that time and hence will not discuss the X-ray signal any further.

If the ultrarelativistic outflow is "on-beam," the optical/radio afterglow is relatively long-lasting and the peak of the optical emission from the forward shock is expected to be within 10²–10³ s after the merger, depending mainly on the initial bulk Lorentz factor of the outflow. The optical afterglow emission of some on-beam GRBs, although missed by gamma-ray detector(s), is expected to be detectable by ZTF and LSST⁶ if the bursts are within the sensitivity distance of the advanced LIGO/VIRGO network (see Figure 2). We do not discuss the radio afterglow from SGRBs and long–short GRBs since it has rarely been detected (see Fong et al. 2015, and references therein). If the GRB outflow is "off-beam" with an angular separation ${\rm{\Delta }}\theta$ , the forward shock emission will not "enter" the line of sight until its bulk Lorentz factor has dropped to $\approx 1/{\rm{\Delta }}\theta$ (Janka et al. 2006). The off-beam timescale is related to the on-beam one by ${{dt}}_{{\rm{off}}}\approx (1+{{\rm{\Gamma }}}^{2}{\rm{\Delta }}{\theta }^{2}){{dt}}_{{\rm{on}}}$ . On the other hand, we have ${\rm{\Gamma }}\approx 7\,{E}_{{\rm{k,51}}}^{1/8}{n}_{-2}^{-1/8}{t}_{{\rm{on,}}\,{\rm{d}}}^{-3/8}$ , where ${E}_{{\rm{k,51}}}$ is the kinetic energy of the GRB outflow in units of ${10}^{51}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ , ${n}_{-2}$ is the number density of the circum-burst medium in units of ${10}^{-2}\,{{\rm{cm}}}^{-3}$ , and ${t}_{{\rm{d}}}$ is the timescale in days. Hence the peak emission time of the "off-beam" relativistic ejecta can be estimated as $t\sim 10\,{\rm{day}}\,{E}_{{\rm{k,51}}}^{1/3}{n}_{-2}^{-1/3}{({\rm{\Delta }}\theta /0.2)}^{8/3}$ . At such a late time, the optical emission from the forward shock is likely (much) dimmer than 22nd magnitude for a source at a distance of ~400 Mpc (see Figure 2), and the prospect of detection is not very promising.

**Figure 2.** The r-band afterglow emission of nearby SGRBs and long GRBs if they took place at a luminosity distance of ~400 Mpc. The initial data are adopted from Covino et al. (2006), Malesani et al. (2007), Ofek et al. (2007), Xu et al. (2009), and Fong et al. (2015).
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The radio emission caused by the sub-relativistic outflow is expected to peak years after the merger (Nakar & Piran 2011), too late to be of interest to us. For the macronova emission, a peak is likely in a few days in the ultraviolet/optical band, while the infrared emission may peak in one or two weeks (see, e.g., Li & Paczyński 1998; Barnes & Kasen 2013; Hotokezaka et al. 2013a, for theoretical predictions; see Jin et al. 2015 for the first observed multi-epoch/band macronova light curve). As shown in Jin et al. (2016), macronova emission is likely to have a very promising prospect of detection and can serve as the ideal electromagnetic signal of merger events. The typical discovery timescale is likely to be ~1–10 days.

We therefore conclude that if the electromagnetic counterparts are prompt GRB emission, on-beam forward shock emission, and macronova emission, then

$\begin{eqnarray*}&&{\rm{\Delta }}{t}_{\mathrm{GW}-\mathrm{ph}}\sim 0.01\mbox{--}1\,{\rm{s}},\,0.01\mbox{--}1\,{\rm{day}},\,1\mbox{--}10\,{\rm{days}},\end{eqnarray*}$

respectively. The expected constraints on are shown in Figure 3. $| \varsigma | \lt {10}^{-18}$ may be achievable in a few years.

4.2. Measuring the GW Velocity with a Simultaneous Test of EEP

In this subsection, we consider a simultaneous constraint on the departure of the GW velocity from the speed of light and the possible degree of violation of EEP with a set of data on GW/GRB association. Within the framework of the parameterized post-Newtonian approximation (PPN), deviations from EEP are described by the parameter γ, which is 1 in general relativity (Will 2014). In general, the Shapiro time delay is calculated from ${t}_{{\rm{gra}}}=-\tfrac{1+\gamma }{{c}^{3}}{\int }_{{r}_{{\rm{o}}}}^{{r}_{{\rm{e}}}}U(r(t);t){dr}$ , where the integration is along the path of the photon emitted form the source at ${r}_{{\rm{e}}}$ and received at ${r}_{{\rm{o}}}$ , and $U(r(t);t)$ is the gravitational potential (Shapiro 1964). If the PPN parameter γ is variable for different species of neutral particles, two kinds of particles emitted simultaneously from the source would arrive at different times, and the corresponding time lag is governed by ${\rm{\Delta }}{t}_{{\rm{gra}}}=-\tfrac{{\rm{\Delta }}\gamma }{{c}^{3}}{\int }_{{r}_{{\rm{o}}}}^{{r}_{{\rm{e}}}}U(r(t);t){dr}$ (Shapiro 1964; Krauss & Tremaine 1988; Longo 1988; Sivaram 1999). In this work we focus on the Shapiro time delay between photons and GWs caused by the gravitational potential of the Milky Way (see Sivaram (1999) for the brief idea and Wu et al. (2016) for a dedicated investigation). Moreover, we adopt the Keplerian potential, for which the Shapiro time delay can be well approximated by (Misner et al. 1973; Longo 1988; Sivaram 1999; Wu et al. 2016)

$\begin{eqnarray}{\rm{\Delta }}{t}_{{\rm{gra}}} & \approx & {\rm{\Delta }}\gamma \displaystyle \frac{{{GM}}_{{\rm{MW}}}}{{c}^{3}}\mathrm{ln}\left(\displaystyle \frac{D}{b}\right)\\ & \approx & 1.7\times {10}^{7}\,{\rm{s}}\,{\rm{\Delta }}\gamma \left(\displaystyle \frac{{M}_{{\rm{MW}}}}{6\times {10}^{11}{M}_{\odot }}\right)\displaystyle \frac{\mathrm{ln}(D/b)}{4\mathrm{ln}10},\end{eqnarray} \tag{ 8 }$

where ${M}_{{\rm{MW}}}\approx 6\times {10}^{11}\,{M}_{\odot }$ is the mass of the Milky Way, D is the distance of the cosmological transient from the Earth, and b is the impact parameter of the particle paths relative to the center of the Milky Way, and we have normalized $\mathrm{ln}(D/b)$ to the value of $4\mathrm{ln}10$ to address the facts that $D\sim 100$ Mpc in the advanced LIGO/VIRGO era and $d\sim 10$ kpc.

As in the "canonical approach," we assume that ${v}_{{\rm{g}}}$ is a constant. Then the observed time delay consists of three parts, i.e.,

$\begin{eqnarray}&&{\rm{\Delta }}{t}_{\mathrm{GW}-\mathrm{ph}}={\rm{\Delta }}{t}_{{\rm{e}}}-\delta {t}_{{\rm{o}}}+{\rm{\Delta }}{t}_{{\rm{gra}}},\end{eqnarray} \tag{ 9 }$

where $\delta {t}_{{\rm{o}}}\approx 2\times {10}^{16}\,{\rm{s}}\,\varsigma (D/200\,{\rm{Mpc}})$ . In the specific model of "Dark Matter Emulators" the GWs are expected to arrive earlier than the simultaneously emitted GRB photons by $\sim {10}^{3}$ days (Desai et al. 2008), while in reasonable astrophysical models the GW signal should precede the GRB for a given source. Hence an almost simultaneous arrival of the GW/GRB signals requires a subluminal GW with a $\varsigma \sim 4\times {10}^{-9}(D/200\,{\rm{Mpc}})$ , which violates the "submuminal constraint" set by ultrahigh-energy cosmic rays (i.e., $0\lt \varsigma \lt 2\times {10}^{-15}$ ) and in turn rules out the Dark Matter Emulators but favors the dark matter model (see also Kahya & Desai 2016 for a discussion of the possible GW150914/GBM transient association).

If two events of association between GWs and electromagnetic counterparts are observed (which are marked by subscripts 1 and 2), we have

$\begin{eqnarray}{\rm{\Delta }}\gamma & = & 5.5\times {10}^{-7}{\left(\displaystyle \frac{{M}_{{\rm{MW}}}}{6\times {10}^{11}{M}_{\odot }}\right)}^{-1}\\ & & \times \displaystyle \frac{{D}_{2}{\rm{\Delta }}{t}_{\mathrm{GW}-\mathrm{ph},1}^{\prime }-{D}_{1}{\rm{\Delta }}{t}_{\mathrm{GW}-\mathrm{ph},2}^{\prime }}{{D}_{2}\mathrm{ln}\left(\tfrac{{D}_{1}}{{b}_{1}}\right)-{D}_{1}\mathrm{ln}\left(\tfrac{{D}_{2}}{{b}_{2}}\right)},\end{eqnarray} \tag{ 10 }$

and

$\begin{eqnarray}&&\varsigma ={10}^{-16}\\ &&\ \times \displaystyle \frac{100\,{\rm{Mpc}}\,\left[\mathrm{ln}\left(\tfrac{{D}_{2}}{{b}_{2}}\right){\rm{\Delta }}{t}_{\mathrm{GW}-\mathrm{ph},1}^{\prime }-\mathrm{ln}\left(\tfrac{{D}_{1}}{{b}_{1}}\right){\rm{\Delta }}{t}_{\mathrm{GW}-\mathrm{ph},2}^{\prime }\right]}{{D}_{2}\mathrm{ln}({D}_{1}/{b}_{1})-{D}_{1}\mathrm{ln}({D}_{2}/{b}_{2})},\end{eqnarray} \tag{ 11 }$

where ${\rm{\Delta }}{t}_{\mathrm{GW}-\mathrm{ph},1}^{\prime }\equiv {\rm{\Delta }}{t}_{\mathrm{GW}-\mathrm{ph},1}-{\rm{\Delta }}{t}_{{\rm{e,1}}}$ and ${\rm{\Delta }}{t}_{\mathrm{GW}-\mathrm{ph},2}^{\prime }\equiv {\rm{\Delta }}{t}_{\mathrm{GW}-\mathrm{ph},2}-{\rm{\Delta }}{t}_{{\rm{e,2}}}.$ (Note that the relationship ${\rm{\Delta }}{t}_{\mathrm{GW}-\mathrm{ph}}\geqslant {\rm{\Delta }}{t}_{{\rm{e}}}$ , as expected in general relativity, may be invalid for the assumptions of the current model.) For a set of GW signals with electromagnetic counterparts, the $({\rm{\Delta }}{t}_{\mathrm{GW}-\mathrm{ph}},\,D,\,b)$ are available and the main uncertainty in constraining and ${\rm{\Delta }}\gamma$ is the accuracy of estimating ${\rm{\Delta }}{t}_{{\rm{e}}}$ . Here we do not simply take ${\rm{\Delta }}{t}_{{\rm{e}}}$ as zero and focus on the events of SGRBs or short–long GRBs with associated GW signals, for which ${\rm{\Delta }}{t}_{{\rm{e}}}$ can be relatively reasonably estimated (see Table 1). In particular, for GRBs from NS–BH mergers we expect that ${\rm{\Delta }}{t}_{{\rm{e}}}\leqslant {T}_{90}$ (unless T₉₀ is significantly shorter than ${\rm{\Delta }}{t}_{{\rm{laun}}}\,(\sim 10\,{\rm{ms}})$ , which has not been recorded by Swift yet).

The conservative constraints on $| {\rm{\Delta }}\gamma |$ and $| \varsigma |$ are

$\begin{eqnarray}&&| {\rm{\Delta }}\gamma | \leqslant 5.5\times {10}^{-7}{\left(\displaystyle \frac{{M}_{{\rm{MW}}}}{6\times {10}^{11}{M}_{\odot }}\right)}^{-1}\\ &&\ \times \displaystyle \frac{| {D}_{2}{\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB},1}-{D}_{1}{\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB},2}| +| {D}_{2}{\rm{\Delta }}{t}_{{\rm{e,1}}}-{D}_{1}{\rm{\Delta }}{t}_{{\rm{e,2}}}| }{| {D}_{2}\mathrm{ln}({D}_{1}/{b}_{1})-{D}_{1}\mathrm{ln}({D}_{2}/{b}_{2})| },\quad \end{eqnarray} \tag{ 12 }$

and

$\begin{eqnarray}&&| \varsigma | \leqslant {10}^{-16}\mathrm{ln}\left(\displaystyle \frac{{D}_{2}}{{b}_{2}}\right)\\ &&\quad \times \displaystyle \frac{| {\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB},1}-{ \mathcal R }{\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB},2}| +| {\rm{\Delta }}{t}_{{\rm{e,1}}}-{ \mathcal R }{\rm{\Delta }}{t}_{{\rm{e,2}}}| }{\left|\tfrac{{D}_{2}}{100\,{\rm{Mpc}}}\mathrm{ln}({D}_{1}{/b}_{1})-\tfrac{{D}_{1}}{100\,{\rm{Mpc}}}\mathrm{ln}({D}_{2}{/b}_{2})\right|},\end{eqnarray} \tag{ 13 }$

respectively, where ${ \mathcal R }\equiv \mathrm{ln}({D}_{1}/{b}_{1})/\mathrm{ln}({D}_{2}/{b}_{2})$ . In future data it may be possible to identify two or more GRBs from NS–BH mergers associated with GW signals, so as a conservative estimate we take into account the fact that $| {\rm{\Delta }}{t}_{{\rm{e,1}}}-{ \mathcal R }{\rm{\Delta }}{t}_{{\rm{e,2}}}| \lt {\rm{\Delta }}{t}_{{\rm{e,1}}}+{ \mathcal R }{\rm{\Delta }}{t}_{{\rm{e,2}}}$ and further replace ${\rm{\Delta }}{t}_{{\rm{e}}}$ by T₉₀. For ${D}_{2}\geqslant 1.5{D}_{1}$ , ${b}_{2}\sim {b}_{1}$ , ${\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB},1}\approx {\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB},2}$ , and ${T}_{\mathrm{90,1}}\approx {T}_{\mathrm{90,2}}$ , the above two constraints (on $| {\rm{\Delta }}\gamma |$ and $| \varsigma |$ ) further reduce to

$\begin{eqnarray}&&| {\rm{\Delta }}\gamma | \lt 6\,\times \,{10}^{-8}\,{\left(\displaystyle \frac{{M}_{{\rm{MW}}}}{6\times {10}^{11}{M}_{\odot }}\right)}^{-1}\displaystyle \frac{{\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB},1}+{T}_{{\rm{90,1}}}}{[\mathrm{ln}({D}_{1}/{b}_{1})/4\mathrm{ln}10]},\end{eqnarray} \tag{ 14 }$

and

$\begin{eqnarray}&&| \varsigma | \lt {10}^{-17}\,\displaystyle \frac{({\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB},1}+{T}_{{\rm{90,1}}})}{[\mathrm{ln}({D}_{1}/{b}_{1})/4\mathrm{ln}10]}\displaystyle \frac{\mathrm{ln}({D}_{2}/{D}_{1})}{[({D}_{2}-{D}_{1})/100\,{\rm{Mpc}}]}.\end{eqnarray} \tag{ 15 }$

Interestingly, such constraints can be as tight as the bounds on ${\rm{\Delta }}\gamma$ or $| \varsigma |$ set by excluding either the EEP test or the departure of ${v}_{{\rm{g}}}$ from c.

5. DISCUSSION

SGRBs are widely believed to be powered by the mergers of compact object binaries. Note that the BH–NS merger rate is generally expected to be $\sim 1/10$ times the NS–NS merger rate (Abadie et al. 2010). Hence most SGRBs are expected to be from NS–NS mergers and a small fraction of events may be due to NS–BH mergers. Though in the upcoming era of the advanced LIGO/VIRGO network the prospect of detecting GW-associated SGRBs is promising, none of the nearby (i.e., $z\lt 0.2$ ) SGRBs are found within the sensitivity distance of the upcoming advanced LIGO/VIRGO network, ${D}_{{\rm{\ast ,BNS}}}\approx 400\,{\rm{Mpc}}\,(9/{\rho }_{\ast })$ for ${\rho }_{\ast }\geqslant 9$ . Such a non-identification, though still understandable (see Equation (3)), is somewhat disappointing. Interestingly, we find that GRB 060505, one supernova-less long event (also known as a long–short GRB), if powered by a NS–NS merger, is likely within the distance of ${D}_{{\rm{\ast ,BNS}}}({\rho }_{\ast }\approx 9)$ . The other long–short burst, GRB 060614, accompanied by a macronova signal that is plausibly powered by a NS–BH merger, is within the distance of ${D}_{* ,\mathrm{NS}\mbox{--}\mathrm{BH}}({\rho }_{* }\approx 9)$ . Therefore in the era of GW astronomy, the origin of some long–short GRBs in the merger of a compact object binary, as favored by the macronova signature displayed in GRB 060614, will be unambiguously tested. We hence suggest that both SGRBs and long–short GRBs are prime targets for the advanced LIGO/VIRGO network, and γ-ray detectors with wide fields of view are encouraged to monitor the sky continually to get accurate information on the properties of the prompt emission.

In the era of the advanced LIGO/VIRGO, reasonably large GRB samples from BNS mergers and from NS–BH mergers can be established (see Section 2). Motivated by such a fact, we have examined the possible distribution of ${\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB}}$ and the relation between T₉₀ and ${\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB}}$ for each sample. As summarized in Table 1, in the case of ${R}_{{\rm{pro}}}\ll {10}^{16}$ cm, which represents the scenario of photospheric radiation and regular (magnetized) internal shock radiation, it is expected that (1) ${\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB}}$ is significantly longer for BNS mergers than for NS–BH mergers, i.e., ${\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB}}(\mathrm{NS}\mbox{--}\mathrm{BH})\ll {\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB}}(\mathrm{NS}\mbox{--}\mathrm{NS});$ (2) for NS–BH mergers, ${\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB}}$ is usually expected to be shorter than T₉₀, i.e., ${\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB}}(\mathrm{NS}\mbox{--}\mathrm{BH})\lt {T}_{90}$ . For ${R}_{{\rm{pro}}}\sim {10}^{16}\,{\rm{cm}}$ , we expect that ${\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB}}$ should be comparable to T₉₀ for both BNS and NS–BH merger-powered SGRBs. The comparison with future real data will be helpful in revealing the physics of the central engine. We would also like to point out that in some specific astrophysical or new physics scenarios, the GW may precede the GRB signal significantly (i.e., ${\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB}}\gt 10$ s). If such large time lags are indeed present in the future data, then a statistical study of the distribution of ${\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB}}\gt 10\,{\rm{s}}$ can distinguish between the astrophysical model (for example the model proposed in Rezzolla & Kumar (2015) for some BNS mergers but not for NS–BH mergers) and the new physics (e.g., the superluminal movement of the GW).

To tightly constrain the difference between the GW velocity and the speed of light, the shorter ${\rm{\Delta }}{t}_{\mathrm{GW}-\mathrm{ph}}$ the better (see Section 4). If the electromagnetic counterpart of a GW signal is a GRB, we have ${\rm{\Delta }}{t}_{\mathrm{GW}-\mathrm{ph}}={\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB}}$ . The shortest ${\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB}}$ is expected for prompt BH formation in NS–NS mergers or NS–BH mergers if the onset of the prompt GRB emission is governed by the photosphere or regular internal shocks (i.e., ${R}_{{\rm{pro}}}\ll {10}^{16}$ cm), in such a case the constraint $| \varsigma | \lt {10}^{-16}$ or even tighter is possible. (If the association of GW150914 and GBM transient 150914 is intrinsic, $| \varsigma | \lt {10}^{-17}$ is inferred, as shown in Li et al. (2016) and Ellis et al. (2016).) With two GW/GRB association events that are expected to be available in the near future we can measure the GW velocity with a simultaneous test of EEP. Intriguingly, very accurate measurements/tests are still achievable in such treatments (see Section 4.2).

We thank the anonymous referee for helpful suggestions, Tsvi Piran for discussions, and Chris Messenger and Imre Bartos for comments. This work was supported in part by National Basic Research Programme of China (No. 2013CB837000 and No. 2014CB845800), NSFC under grants 11525313 (i.e., Funds for Distinguished Young Scholars), 11361140349, 11273063, and 11433009, the Foundation for Distinguished Young Scholars of Jiangsu Province, China (Grant No. BK2012047), and the Strategic Priority Research Program (Grant No. XDB09000000).

GRB/GW ASSOCIATION: LONG–SHORT GRB CANDIDATES, TIME LAG, MEASURING GRAVITATIONAL WAVE VELOCITY, AND TESTING EINSTEIN'S EQUIVALENCE PRINCIPLE

Abstract

1. INTRODUCTION

2. THE PROSPECTS OF DETECTING GW SIGNALS ASSOCIATED SGRBs AND LONG–SHORT GRBs

2.1. The Prospect of Detecting SGRBs Associated with GW Signals

2.1.1. Binary NS (BNS) Mergers

2.1.2. BH–NS Mergers

2.2. "Supernova-less" GRBs within the Sensitivity Distance of the Advanced LIGO/VIRGO Network

3. MODEL-DEPENDENT ESTIMATES OF TIME LAG BETWEEN BINARY COALESCENCE AND GRB ONSET

3.1. ${\rm{\Delta }}{t}_{{\rm{laun}}}$ Expected in Different Merger Scenarios

3.1.1. NS–NS Mergers

3.1.2. NS–BH Mergers

3.2. ${\rm{\Delta }}{t}_{{\rm{em}}}$ Expected in Models of Baryonic and Magnetic Outflow

3.2.1. The Bayronic Outflow

3.2.2. The Magnetic Outflow

3.3. Expected Relationship between ${\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB}}$ and T₉₀

4. MEASURING THE VELOCITY OF THE GRAVITATIONAL WAVE AND TESTING EINSTEIN'S EQUIVALENCE PRINCIPLE (EEP)

4.1. Measuring the GW Velocity: the "Canonical" Approach

4.2. Measuring the GW Velocity with a Simultaneous Test of EEP

5. DISCUSSION

Footnotes

References

GRB/GW ASSOCIATION: LONG–SHORT GRB CANDIDATES, TIME LAG, MEASURING GRAVITATIONAL WAVE VELOCITY, AND TESTING EINSTEIN'S EQUIVALENCE PRINCIPLE

Abstract

1. INTRODUCTION

2. THE PROSPECTS OF DETECTING GW SIGNALS ASSOCIATED SGRBs AND LONG–SHORT GRBs

2.1. The Prospect of Detecting SGRBs Associated with GW Signals

2.1.1. Binary NS (BNS) Mergers

2.1.2. BH–NS Mergers

2.2. "Supernova-less" GRBs within the Sensitivity Distance of the Advanced LIGO/VIRGO Network

3. MODEL-DEPENDENT ESTIMATES OF TIME LAG BETWEEN BINARY COALESCENCE AND GRB ONSET

3.1. {\rm{\Delta }}{t}_{{\rm{laun}}} Expected in Different Merger Scenarios

3.1.1. NS–NS Mergers

3.1.2. NS–BH Mergers

3.2. {\rm{\Delta }}{t}_{{\rm{em}}} Expected in Models of Baryonic and Magnetic Outflow

3.2.1. The Bayronic Outflow

3.2.2. The Magnetic Outflow

3.3. Expected Relationship between {\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB}} and T90

4. MEASURING THE VELOCITY OF THE GRAVITATIONAL WAVE AND TESTING EINSTEIN'S EQUIVALENCE PRINCIPLE (EEP)

4.1. Measuring the GW Velocity: the "Canonical" Approach

4.2. Measuring the GW Velocity with a Simultaneous Test of EEP

5. DISCUSSION

Footnotes

References

3.1. ${\rm{\Delta }}{t}_{{\rm{laun}}}$ Expected in Different Merger Scenarios

3.2. ${\rm{\Delta }}{t}_{{\rm{em}}}$ Expected in Models of Baryonic and Magnetic Outflow

3.3. Expected Relationship between ${\rm{\Delta }}{t}_{\mathrm{GW} \mbox{-} \mathrm{GRB}}$ and T₉₀